#7712577

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#37554	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4500 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

(2,5)(6,8)(12,19)(16,21)(17,23)(22,24)(25,26)(27,31)(28,32)(29,36)(30,37)(33,39)(34,38)(35,40);(1,2,4,5)(3,6,7,8)(9,10)(11,15)(12,16)(13,14)(17,22)(18,20)(19,21)(23,24)(26,27)(32,35)(33,36)(34,37);(1,3,4,7)(2,6,5,8)(10,14)(15,20)(16,21)(22,24)(25,27,31,26)(28,32,40,35)(29,33,39,36)(30,37,38,34);(9,11,13,18)(10,15,14,20)(12,17,19,23)(16,22,21,24)(25,28)(26,32)(27,35)(29,38)(30,39)(31,40)(33,37)(34,36);(9,10)(11,15)(12,16)(13,14)(17,22)(18,20)(19,21)(23,24)(25,29,31,39)(26,33,27,36)(28,38,40,30)(32,37,35,34);(9,12,13,19)(10,16,14,21)(11,17,18,23)(15,22,20,24)(25,30,31,38)(26,34,27,37)(28,39,40,29)(32,36,35,33)

Polycyclic presentation:

2^9|b^2=gh,c^2=g,d^2=hi,e^2=h,f^2=i,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^c=dh,e^c=ei

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7619773);